This paper adresses the numerical modelling of the solidification of a binary alloy which obeys a liquidus-solidus phase diagram. In order to capture the moving melting front, we introduce a Lagrange projection scheme based a random sampling projection. Using a finite volume formulation, we define accurate numerical fluxes for the temperature and concentration fields which guarantee the sharp treatment of the boundary conditions at the moving front, especially the jump of the concentration according to the liquidus-solidus diagram. We provide some numerical illustrations which assess the good behaviour of the method: maximum principle, stability under CLF condition, numerical convergence toward self-similar solutions, ability to handle two melting fronts
